Development of Load and Resistance Factors for
Reinforced Concrete Structural Members in North
Cyprus
Abdulhamid Sagir Mahmud
Submitted to the
Institute of Graduate Studies and Research
in partial fulfilment of the requirements for the degree of
Master of Science
in
Civil Engineering
Eastern Mediterranean University
July 2017
Approval of the Institute of Graduate Studies and Research
______________________________ Prof. Dr. Mustafa Tümer
Director
I certify that this thesis satisfies the requirements as a thesis for the degree of Master of Science in Civil Engineering.
__________________________________ Assoc. Prof. Dr. Serhan Şensoy Chair, Department of Civil Engineering
We certify that we have read this thesis and that in our opinion, it is fully adequate in scope and quality as a thesis for the degree of Master of Science in Civil Engineering.
__________________________________ Assoc. Prof. Dr.Serhan Şensoy
Supervisor
Examining Committee 1. Assoc. Prof. Dr. Mehmet Cemal Geneş ___________________________
2. Assoc. Prof. Dr. Giray Özay ____________________________ 3. Assoc. Prof. Dr. Serhan Şensoy ____________________________
iii
ABSTRACT
Load and Resistance Factor Design (LRFD), is a widely used procedure in the design of reinforced concrete, wood and steel structures. It is a reliability-based procedure for design, which gives a framework that is consistent with civil engineering design codes, in accordance with reliability theories. In this study, Advance First Order Second Moment (AFOSM) approach is used as the reliability approach in carrying out the analysis. Uncertainties related to material properties (i.e. compressive strength of concrete, yield and ultimate strength of reinforcing steel bars.), dimensions of reinforced concrete structural members (beams and columns) and the effect of load variables (i.e. Dead and Live load), are considered. Under the framework of AFOSM the failure mode in different reinforced concrete structural members were analyzed, which focused mainly on flexure failure, shear failure and the combined action of flexure and axial load failure. Reliability indexes are calculated according to the flexure and shear failure modes in beams and columns, in addition to failure due to the combine action of flexure and axial load on columns.
Target reliability indexes are selected for different load combinations from values reported by other researchers from different countries, which are used as the safety level to evaluate the computed reliability indexes. New load and resistance factors are selected for different failure modes in different structural members, considering the design practice in North Cyprus and specifications given in the Turkish codes (e.g. TS500).
iv
ÖZ
Yük ve Dayanım Katsayılarına göre Tasarım (YDKT), betonarme, ahşap ve çelik yapı elemanlarının tasarımında yaygın olarak kullanılan bir yöntemdir. Bu yöntem güvenilirlik esasına dayanan, tasarım yönetmelikleri çerçevesinde ve güvenilirlik teorilerine uygun bir yöntemdir. Bu çalışmada analizler “Geiştirilmiş Birinci Mertebe İkinci Moment” (GBMİM) yaklaşımı ile yapılmıştır. Çalışma kapsamında malzeme özellikleri (beton basınç mukavemeti, beton çelik çubukları çekme dayanımı v.b), betonarme elemanların boyutları (kirişier ve kolonlar) ve yük değişkenlerinin etkisine (sabit ve hareketli yükler) ilişkin belirsizlikler dikkate alınmıştır. GBMİM yaklaşımı çerçevesinde farklı betonarme yapı elemeanlarının göçme durumu incelenmiştir. Bu çalışmada esas olarak eğilme, kesme ve eksenel kuvvet-eğilme etkileşimindeki elemanlar dikkate alınmıştır. Bu bağlamda betonarme kirişlerde eğilme ve kesme göçme durumları ile eksenel kuvvet ve eğilme etkisindeki betonarme kolonların “güvenilirlik indeksleri” hesaplanmıştır.
Farklı yük birleşimlerine göre daha önceki çalışmalarda farklı ülkeler için rapor edilen hedef güvenilirlik indeksi değerleri seçilmiştir. Hesaplanan güvenilirlik indeksi seçilen hedef güvenilirlik indeksi ile kıyaslanarak yeterli güvenlik seviyesine ulaşılmıştır. Yapılan çalışma kapsamında Kuzey Kıbrıs’ta tasarım uygulamaları ve TS500’de belirtilen kurallar çerçevesinde yeni yük ve dayanım katsayıları belirlenmiştir.
Anahtar Kelimeler: Belirsizlik Modeli, Güvenilirlik, YDKT, Güvenilirlik İndeksi,
v
ACKNOWLEDGEMENT
I would like to express my heartfelt gratitude to my supervisor Assoc. Prof. Dr. Serhan Şensoy for his valuable suggestions, supports, encouragement, criticism, guidance and the many hours that he devoted to discussing my study.
I am indebted to Prof. Dr. M. Semih Yücemen who not only helped me with my study, but also gave me the basics of reliability based design as a graduate course.
I owe special thanks to Prof. Dr. Özgür Eren for giving the opportunity to further my education. I am grateful for his encouragement and support, throughout my graduate studies.
Special thanks to material laboratories of Mechanical and Civil engineering department of EMU for providing the data used in this study.
Finally, my deepest gratitude goes to my family for their ongoing love, support and prayers. My beloved mother, who has always been encouraging and supportive of my entire studies. My precious sister, who always prayed for my success. My amazing brother, who always encourage and advise me.
vi
TABLES OF CONTENTS
ABSTRACT ... iii
ÖZ ... iv
ACKNOWLEDGEMENT ... v
LIST OF TABLES ... viii
LIST OF FIGURES ... ix
LIST OF SYMBOLS ... x
1 INTRODUCTION ... 1
1.1 Overview ... 1
1.2 Literature Review ... 2
1.3 Aim of this Study ... 5
1.4 Outline of this Study ... 6
2 RELIABILITY THEORY ... 8
2.1 Introduction ... 8
2.2 Uncertainty Modeling ... 9
2.3 The Reliability Index ... 11
2.4 Computing the Reliability Index ... 12
3 UNCERTAINTY QUANTIFICATION OF THE RESISTANCE VARIABLES .. 18
3.1 Introduction ... 18
3.2 Concrete ... 18
3.3 Reinforcing Steel Bars ... 28
3.4 Dimensions ... 34
3.5. Area of Reinforcing Steel Bars... 37 4 ANALYSIS OF THE RESISTANCE PARAMETER CONSIDERING
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DIFFERENT FAILURE MODES. ... 38
4.1 Introduction ... 38
4.2 Failure Modes in Reinforced Concrete Beams ... 38
4.3 Failure Modes in Reinforced Concrete Columns ... 43
5 LOAD VARIABLES ... 47
5.1 Introduction ... 47
5.2 Dead Load ... 48
5.3 Live Load ... 49
6 CALIBRATION AND SELECTION OF LOAD AND RESISTANCE FACTORS ... 50
6.1 Introduction ... 50
6.2 Load Parameters ... 51
6.3 Resistance Parameters ... 53
6.4 Selection of Target Reliability Index ... 54
6.5 Reliability Analysis for each Failure Mode. ... 55
6.6 Reliability Index and Selection of Load and Resistance Factors ... 58
7 CONCLUSION ... 61
viii
LIST OF TABLES
Table 1. 28 day Statistical parameters on the compressive strength data ... 24
Table 2. Reinforcing steel bars required standards according to different countries (from Firat, 2007) ... 29
Table 3. Analysis on yield strength for 420(a) reinforce steel bars data... 30
Table 4. Analysis on Ultimate strength of 420(a) reinforcing steel bars data ... 30
Table 5. Statistical analysis results on column dimensions (Firat, 2007) ... 36
Table 6. Statistical analysis results on column dimensions (Firat, 2007) ... 37
Table 7. Uncertainty analysis on dead load (from Firat, 2007) ... 48
Table 8. Uncertainty analysis on live load ... 49
Table 9. Statistical analysis on dead load and live load ... 53
Table 10. Target reliability indexes for different load combinations and different structural members according to different studies ... 54
Table 11. Resistance factors and recommended load combination for beam in flexure failure mode ... 58
Table 12. Resistance factors and recommended load combination for beam in flexure failure mode ... 59
Table 13. Resistance factors and recommended load combination for column in combined action failure mode ... 59
Table 14. Resistance factors and recommended load combination for structural members in all failure modes ... 60
ix
LIST OF FIGURES
Figure 1. Definition of β using the Safety Margin ... 12
Figure 2. Hasofer-Lind Reliability Index β2 ... 15
Figure 3. Compressive strength distribution for C16 data ... 21
Figure 4. Compressive strength distribution for C20 data. ... 21
Figure 5. Compressive strength distribution for C25 data ... 22
Figure 6. Compressive strength distribution for C30 data ... 22
Figure 7. Compressive strength distribution for C30 data ... 23
x
LIST OF SYMBOLS
A Cross sectional area A Peak ground acceleration AS Reinforcement area
ASW Cross sectional area of stirrups
b Member width
c Distance from the neutral axis to outer compressive fiber in a T cross-section
cb Depth of neutral axis at the balanced case in reinforced concrete cross section
D Dead load effect Df Failure domain DS Safety domain
d Depth of the member (db for beam, dc for column) de Effective depth of the member
FX,fx Cumulative distribution function (CDF) and probability density function of variable X, respectively.
fc Concrete compressive strength fct Tensile strength of concrete fS Steel stress
fy Yield strength of steel bars
fyw Yield strength of shear reinforcement
k1 A dimensionless coefficient which is a function of strength of concrete
xi
L Live load effect of maximum live load
M Safety margin
Mr Bending moment capacity N Correction factor
N Axial load
Pf Probability of failure P(E) Probability of event E
PS Survival probability (reliability) R Generalized resistance
R Rate of loading s Spacing of stirrups
t Depth of the flange thickness U Effect of factored load V Total design base shear Vc Shear resistance of concrete Vd Maximum design shear force Vr Shear strength
Vw Resistance of shear reinforcement W Width (Wb for beam, Wc for column) X Basic random variable
X Mean of X
X′ Nominal of X
X∗ Design value of X
X� The model used to estimate X Xapt Arbitrary point-in-time value of X
xii
xi Distance between neutral axis and ith steel layer in reinforced
concrete cross section β Reliability index βT Target reliability index 𝛾 Generalized load factor ∆ Prediction uncertainty δ Basic variability
εcu Ultimate strain in concrete
µ Mean value
ρ Steel ratio
ρb Balanced steel ratio
ρ′ Compression reinforcement ratio σ Standard deviation
σS Steel stress
Ω Total variability
ϕ Generalized resistance factor
AFOSM Advanced First Order Second Moment Method APT Arbitrary point-in-time
a.p.t. Arbitrary point-in-time
C.D.F. Cumulative distribution function c.o.v. Coefficient of variation
FOSM First Order Second Moment Method JCSS Joint Committee on Structural Safety LRFD Load and Resistance Factor Design RC Reinforced Concrete
xiii
TEC Turkish Earthquake Code (Specification for Structures to be Built in Earthquake Areas)
1
Chapter 1
INTRODUCTION
1.1 Overview
In the design of structures the main priority is to have a structure that is both safe, serviceable and at the same time economical, this is the basis for every structural engineer when it comes to the design of any structure. Uncertainty of any kind that might be resulting from inadequate information, prediction error and at times human error, should be considered when an engineer is trying to make a design decision. Safety requirements are introduced in engineering designs to account for the risks associated with these uncertainties.
The load and resistant factors design (LRFD) accounts for uncertainty related the parameters of load and resistance by the combination of limit state design (LSD) and probabilistic approach. The limit state design is divided into two categories, serviceability limit state (SLS) and ultimate limit state (ULS) (Salgado, 2008). The SLS is related to the malfunctioning of structures, for example the differential or uniform settlement in structures, while the ULS is associated with lack of safety in structures, which includes failure or collapse of structures. ULS occurs when the load is equal to the resistance of a structural system, when this happens the system fails. Therefore, for a good engineering design to be successful the ULS has to be identified and prevented.
2
LRFD tries to keep the level of probability of failure from exceeding the allowable safety level for safety purposes (i.e. at target reliability). The LSD frame work is used to explain LRFD, by analyzing the ULS using some partial factors on resistance and load. The partial factors related to the load and the resistance are computed based on their uncertainties.
Design codes are created to provide a safe and economical guide in engineering designs. Design codes provides a probabilistic approach where design is concerned, this is due to the short comings in deterministic approach of solving structural safety problems. Since the design of structures has to be done in the presence of some uncertainties, probabilistic approach is used to quantify those uncertainties for safety purposes.
The reliability of a structure is determined by comparing the load effect to the resistance effect. In probabilistic approach load and resistance parameters are treated as random variables and the safety is determined by using a tolerable reliability index. The reliability of a whole structure is the sum of the reliabilities of individual structural members. In this approach, the ratio of mean to nominal value and the total uncertainties is computed for the purpose of calibration.
1.2 Literature Review
The concepts of probability assessment on structural safety was first introduced in the beginning of the 20th century. In the early 60s of that century, the American Concrete Institute (ACI) building code introduced a design that is based on the ultimate limit state, which used load factors to increase the load and strength reduction factors to reduce the strength, this design approach is known as the
3
ultimate strengths design (USD). All studies related to this field started in the late 60s and an increase in interest on this structural reliability topics has been shown ever since then.
In the year 1979, Drs. Cornell, Galambos, Ellingwood and MacGregor all came together in the center for building technology at the National Bureau of Standards, which is currently knows as National Institute of Standards and technology with that purpose of recommending some set of universal load and resistance factors, these factors will be utilized during the design stage of structures. The meeting of the above Drs lead to several outcomes that was published in different papers some examples are;
Ellingwood et al in 1980. Galambos et al in 1982. Ellingwood et al in 1982.
The results of that meeting lead to the development of the fundamental sets of load and resistance factors that were amalgamated in the 1982 ANSI A58.1.
After the fundamental factors for the load and resistance were created, further research continued in order to develop the load and resistance for different regions. Rackwitz (2000), in his study used the total cost minimization as a decision model to assess the target reliability in the process of developing a design code, the cost he considered involves both the maintenance and the reconstruction cost. A joint committee on structural safety (JCSS) was created by a Liaison committee that is responsible with the coordination of the activities of the six international association in civil engineering. The aim of creating the committee is to improve the knowledge
4
related to structural safety, which lead to the development of a code in 2001, which was based on probabilistic design but there was no consideration of some information that includes the mechanical models such as the shear capacity, buckling and flexural failure.
The calibration of the Building code Requirement for Structural Concrete (ACI 318), was conducted based on the study of Nowak & Szerszen (2003a and b). The study was done in two parts ‘a’ and ‘b’, the first publication discussed the topic of statistical model for the resistance parameters, which creates the basics of the selection of the resistance factors (strength reduction factors). The second publication discussed reliability analysis and methods of selecting a resistance factor.
Design codes and specifications are not available for North Cyprus, hence the Turkish design code (TS500) is adopted. Thus research in this topic is not readily available here. The Turkish design codes and specifications are used as the guide for design here in North Cyprus, the design code follows a deterministic approach and reliability based design were not implemented during its calibration. Yücemen and Gulkan (1989), were the first to start any significant research on this topic in Turkey. Their study was based on suggesting some sets of reliability based load and resistance factors related to the design of reinforced concrete beams. Kömürcü (1995) followed up with a research on developing a reliability based design using the local conditions and the design practice in Turkey for reinforced concrete beams considering the flexural failure. Later, Firat (2007) in his study developed a reliability based design criterion for reinforced concrete beams, columns and shear walls, considering different failure modes in accordance to the conditions and design practices in Turkey, which is presently the most updated study in this field in Turkey.
5
In this study, to a certain extent is a follow up on the research conducted by Firat (2007), here the local data and the condition regarding the resistance variables of North Cyprus were used, also the load variable were selected and compared for different cities and with the results of other international researchers. The use of the local conditions in North Cyprus in determining appropriate load factors is the main objective of this study.
1.3 Aim of this Study
The aim of this study is to develop a reliability based load and resistance factors design criterion for reinforced concrete structural members considering the local conditions in North Cyprus using a probabilistic approach. For the purposes of this study, load and resistance parameters are treated as random variables. Local data used for the evaluation of these parameters were collected in North Cyprus and from values reported in international literature.
The Advance First Order Second Moment Method (AFOSM) is the approach adopted as the structural reliability model to propose the new sets of load and resistance factors. The effects of the loads on the structure coming from dead load and live load are estimated and also the resistance parameters that include reinforced concrete beams and columns in flexure and shear failure mode, together with the combine action of flexure and axial load on the column. The ratio of mean to nominal values is computed for both the load and resistance parameters solely for the purpose of calibration.
6
The Turkish design code and specifications is used as the guide in this study and the local conditions in North Cyprus are used to quantify the uncertainties related to the resistance and load parameters.
1.4 Outline of this Study
This study follows the development of LRFD based design procedure within the framework of LSD for reinforced concrete structural members. Chapter 1 gives an overview and a literature survey of work that has been done in this field. Chapter 2 give the background of the structural reliability models and probabilistic approach used in the analysis of uncertainties, it goes on to give definitions of basic headings such as reliability index and methods of computing it. Then in the conclusion part of the chapter it states the method selected for this study and the reason of this selection.
Chapter 3 is devoted on computing the ratio of mean to nominal values and the total uncertainty in the resistance variables, the assessment of local data on yield strength of reinforcing steel bars, dimension of structural members and the compressive strength of concrete were used in order to quantify the uncertainties in the resistance variables.
Chapter 4 uses the values determined in the analysis of the resistance variables from the previous chapter to compute the value of total uncertainties and the ratio of mean to nominal values in reinforced concrete structural members such as beams and columns for different failure modes which includes; flexure and shear failure mode and the combined action of flexure and axial load, all within the structural reliability framework.
7
In chapter 5, modeling loads used basically for design are analyzed with that help of local data and reports in international literature and those in Turkey. In chapter 6, the resistance criterion for different failure mode of the structural members are evaluated in compliance with reliability based design, target reliability index are selected. Then using the target reliability index a new set of resistance and load factors are selected for the different failure modes in the reinforced concrete structural members. Chapter 7 gives a summary of all content of the report, conclusions on results are also given in this chapter.
8
Chapter 2
RELIABILITY THEORY
2.1 Introduction
Serviceability and safety are important millstones required by an engineering design to ensure that the structural performance is in conformance with the life time design expectations.
A reliability analysis method gives a theoretical frame work used to quantify uncertainties in order to make a better design decision. It aims at evaluating the ability or capability of a system to operate under safety conditions throughout the structure’s life cycle. Computing the probability of failure and reliability index helps in quantifying the risk involved and therefore providing the possible consequences if failure should occur. Problems related to reliability methods are modeled as random variables, a random vector is a group of random variables denoted by X, where 𝑓𝑥(𝑋) is called the joint probability density function (PDF).
A structure is said to be safe when the strength (R), is able to resist the maximum load effect (S), acting on the structure. In reliability theory failure in a member occurs when the strength of the member is less than the load applied on it (i.e. R< S), or the safety margin (M) is less than zero (i.e. M < 0, where M = R- S). The failure probability can be computed by the use of the Eq. (2.1).
9
Where; Pf is the probability of failure and P is the probability that the events in the
brackets will occur.
The failure probability where R and S are independent and normally distributed given that R� and S� are the mean and σR and σS denotes standard deviation of the strength and load effect, respectively, is computed with the following relationship.
Pf= 1 − ϕ �M�σ� = 1 − ϕ � R�− S� �σR2+ σ
S
2� (2.2)
Where ϕ denotes the probability distribution of the standard normal variate.
2.2 Uncertainty Modeling
Uncertainty is said to be involved in almost everything man does, including the structural design and other things that has to do with decision making. (Bulleit, 2008) categorized uncertainties into two;
Aleatory Uncertainty, Epistemic Uncertainty.
Aleatory Uncertainty is referred to as the inherent or natural variability, it occurs due to randomness inherent in nature.
Epistemic Uncertainty occurs due to inadequate information or knowledge, an increase in information and data can be used to reduce this type of uncertainty.
Melchers (1999) further classified uncertainties into different types and how they affect the structural design and its performance. These classifications include;
Phenomenological Uncertainty, Decision Uncertainty,
10 Modeling Uncertainty, Prediction Uncertainty, Physical Uncertainty, Statistical Uncertainty, Human Error.
The uncertainties that are normally considered for the purpose of design are mainly physical uncertainty, statistical uncertainty and model uncertainty. Physical uncertainties mainly involve uncertainties that are associated with the type of loading environment, the geometry of a structure and material properties used. The statistical uncertainties occur due to incomplete information e.g. the number of sample needed for the test is not adequate. Finally, model uncertainties have to be considered for the purpose of accounting for those uncertainties related to the mathematical descriptions that are used to approximate the real physical behavior of the structure.
In order to model the various sources of uncertainties the prediction error and the inherent variability is combined to find the total uncertainty involved in the structural member. According to the formulation given by the First Order Second Moment (FOSM) method, the total uncertainty can be computed by the relationship in Eq. (2.3) below;
ΩXi = �δX2i+ ∆X2i (2.3)
Here ∆Xidenotes the coefficient of variation for the effect related to the epistemic uncertainty (Ni).
δ
x denotes the coefficient of variation related to the inherent11
The inherent variability (Χi), is computed by the relationship suggested by (Ang &
Tang, 1984):
Xi = NiX�i (2.4)
Here Ni is a random factor used for the correction of the uncertainty value; X�i is to
estimate the value of Xi.
The epistemic uncertainty (Ni) is computed by first computing all the possible sources that can cause this type uncertainty and labeling them accordingly to give a sum total of the correction value to be used (i.e. Ni =Ni1 Ni2,..., Nin). According to the
formulation given in FOSM, the epistemic uncertainty can be computed with the following relationship given in Eq. (2.5) and Eq. (2.6). (Ellingwood et al., 1980)
Ni = Ni1Ni2, … … , Nin (2.5)
∆Xi= �∆X2i1+ ∆X2i2+, … . . , + ∆X2in (2.6)
2.3 The Reliability Index
The reliability index is denoted with the symbol β. It is often used as a substitute for the probability of failure. β may be used to compare different structures and can be used as the target in reliability-based design without mentioning a specific probability of failure.
Cornell (1969) defined the reliability index, or safety index, as the mean of the safety margin which is divided by the standard deviation of the safety margin. This formulation of the reliability index is referred to as the Cornell reliability index.
β = M
12
In this formulation β can be interpreted as the distance between the mean of the safety margin and the point of failure, measured in terms of the standard deviation of the safety margin. However, for all definitions, the basic concept of β remains the same; it is a measure of the distance between the most likely state of the structure (mean) and the most likely failure point, in terms of the variation.
Figure 1. Definition of β using the Safety Margin
2.4 Computing the Reliability Index
There are numerous ways to compute the reliability indexes, but here the most popular technique are briefly discussed based on literature.
2.4.1 First Order Reliability Method (FORM)
First Order Reliability (FORM) deals with the first moment and second moment random variables. The procedure entails of two approaches which includes;
First Order Second Moment (FOSM) approach,
Advanced First Order Second Moment (AFOSM) approach.
The distributional information in AFOSM is appropriately used, while in the FOSM approach, distributional information on the random variables is ignored.
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2.4.1.1 First Order Second Moment (FOSM)
First Order Second Moment (FOSM) approach uses the mean value and the standard deviation of the random variables. The linear function of the performance at the mean of the random variables is needed in this method. The linearization of the performance function at the mean of the random variables utilizes the first order Taylor series method of approximation.
Cornell’s reliability index is extended into the formulation used to compute the reliability index. β is defined as the mean of the safety margin divided by the standard deviation of the safety margin. Therefore, to compute the reliability index the mean and standard deviation of the design variables are required. However, to give space for non-linear limit state functions, the mean and standard deviation of the safety margin are computed by using the Taylor series expansion to linearize the safety margin. The value computed for β depends on the point that is chosen to linearize the limit state function. A common choice is the point where each random variable takes on its mean value, resulting in the mean-value, first-order, second-moment reliability index, even though this method is very simple, it has several shortcomings. The most significant shortcoming is that the value of β is not invariant with respect to the limit state functions. For example two mechanically equivalent functions of the same limit state can produce different result for the reliability index.
2.4.1.2 Advanced First Order Second Moment (AFOSM)
Advance First Order Second Moment (AFOSM) approach, can be referred to as ‘Hasofer Lind’ approach of computing β. When all variables and limit state function are transferred to a standard Normal space, the design point is computed by selecting a minimization procedure that gives a mark on the limit state surface and gives the
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minimum distance to the origin. When the reliability index is computed in this procedure is referred to as βHL, the Hasofer Lind reliability index (Hasofer & Lind,
1974). This measure of reliability is invariant with respect to the limit state function, but it utilizes only the information given by the second-moment about the variables, it can not be compared because the procedure does not depend on the curvature of the limit state function at the design point.
In this approach, the reliability index is assessed mainly by transforming the problem to a standard coordinate system. Therefore, the random variable Xi is transformed to
Zi with the relationship given in Eq. (2.8) below:
Zi = Xiσ− Xi
Xi , 𝑖 = 1,2, … . , 𝑛 (2.8)
Where Zi = random variable with mean equal to zero and a unit standard deviation.
Hence, the written equation above can be used to transform an initial limit state surface that is given by g(X) = 0 into a reduced form of the limit state surface given by g(Z) = 0. Therefore, X stands for the ‘original coordinate system while Z represents the ‘transformed or reduced form of the coordinate system’. Due to shortcomings in the FOSM method, (Hasofer and Lind, 1974) proposed a new reliability index denoted by βHL. The approach used in calculating βHL is referred to
as the Advanced First Order Second Moment (AFOSM) approach. βHL, can be
defined as the minimum distance measured from the origin to the failure surface on the Z coordinate. A design point can be defined as any point yielding the shortest distance from the origin to any point on the failure surface.
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Figure 2. Hasofer-Lind Reliability Index βHL
Figure 2 gives a graphical representation of the Hasofer-Lind reliability index βHL,
which is valid only in the case where two variables are involved.
When a nonlinear failure surface is involved, the seeking of a design point and the computation of a reliability index is done by the use of an iterative approach. This is done by first computing the values of Zi’s using Eq. (2.8) and substituting the value
into Eq. (2.9).
Zi∗ = αiβ2 (2.9)
The value of ᾳi’s are the directional cosines that are used in the minimization of the
value of β2. The relationship given in Eq. (2.10) is used to compute the value of ᾳi;
αi = − ∂g/∂Zi �∑�∂g/∂Zi�2�
1/2, i = 1,2, … . , n (2.10)
The iterations are made until the values start to converge in the value of β2 then the
probability of failure can be computed with the formulation given below.
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2.4.2 Monte Carlo Simulation (MCS)
Monte Carlo Simulation (MCS) is a technique with different applications to many different problems. In general the Monte Carlo approach involves the use of a random sampling technique to generate a set of data to be used in the analysis. In application to structural reliability, a random value is generated for each design variable based on the type of distribution that variable follows. These random variables are utilized in order to analyze the limit state equation. For a limit state function less than zero, can be interpreted that the structure has “failed”.
This procedure is iterated several times, the probability of failure is computed by dividing the number of samples that failed by the total amount of simulations. This procedure is very tough and it can be applied in almost any type of limit state function. Thus, the reliability of this procedure lean on the amount of simulations made, and for probabilities of failure that are small the required time for the computation can also be very ambitious. With additional knowledge about the failure region, variance reduction techniques, that include importance sampling, is used to concentrate the simulations in the region of interest and reduce the necessary amount of simulations required to compute the reliability.
2.4.3 Reliability Analysis Method Used Within the Scope of this Study
The Advance First order reliability approach (AFOSM) is the method selected for the course of this study. This procedure was chosen for several reasons, they are;
It is easy to compute,
It considers the short comings of FOSM,
It gives room for many variables without overlapping,
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Though the intent of this study is to calibrate based on component reliability, the limit state chosen for explicit consideration herein is the flexural and shear limit state function, which is used for the flexural and shear failure of a reinforced concrete structural members considering different failure modes. Hence, to make use of other reliability methods would require assessing the reliability against all the different failure modes separately and using principles of system reliability to calculate the reliability. AFOSM can consider all the failure modes simultaneously based on the particular random variables provide during the computation and provide the total reliability against the flexural and shear failure.
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Chapter 3
UNCERTAINTY QUANTIFICATION OF THE
RESISTANCE VARIABLES
3.1 Introduction
In this chapter uncertainties related to the resistance variables are quantified, they include;
Concrete,
Reinforcing Steel Bars, Dimensions,
Area of Reinforcing Steel Bars.
Analyses were carried out using local data and design condition present in North Cyprus. The mean to nominal ratio and the total uncertainties in each resistance variable is computed for the purpose of calibration
3.2 Concrete
Concrete is defined as a mixture of aggregates and paste, which the paste consists of cement and water, the aggregates consists on of sand, gravel and crushed stones. For the purpose of safety in reinforced concrete structures the quality of concrete is important, the quality of concrete can be measured by considering it’s workability, compressive strength, durability, performance, and setting time.
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The use of statistical analysis on the data obtained from concrete test is a more efficient way of evaluating the quality of concrete, a good quality concrete should result in a high mean value and a low coefficient of variation (c.o.v). The variation in concrete quality starts with variation in the properties of the materials used during the mixing stage, these variations in material properties of concrete can be due to the existence of some factors that are present at the mixing stage, such as;
Temperatures. Methods of mixing. Mixing proportion.
In order to produce a high quality concrete careful observations should be made on factors that lead to the production of low quality concrete, these factors include:
Lack of proper control. Lack of supervision.
Not paying attention to details. Poor workmanship.
Most importantly compressive strength is the most affected property of concrete in terms of its mechanical properties, so to control the quality of concrete the compressive strength test is examined on cubic specimens (Ergün & Kürklü, 2012).
3.2.1 Data Evaluation and Analysis
Data were collected from the laboratory of Civil Engineering Department of Eastern Mediterranean University (EMU) North Cyprus, on the compressive strength of cubic concrete specimens with dimension of 150 x 150 x 150 mm. The tests were conducted accordance with the specifications given in TS 3114, samples to be used for this test were cured in the laboratory for the number of days required before
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testing, stored in accordance with the provisions given in TS 3068. A total of 5705 samples were test within the period of 8 years (i.e. 2004 to 2014). The samples of the specimens were obtained from firms around North Cyprus.
The concrete specimens test were conducted for different concrete age ranging from 3 to 55 days compressive strength, for the purpose of this study a 28-day compressive strength is considered since it’s the fundamental measure on the regulations for the strength of concrete. With regards to this regulation samples that were tested below and above the 28-day age, a 28-day compressive strength was approximately computed using a non-linear regressional analysis by utilizing the logarithmic best fit for the data which yielded a mathematical equations suggested in a study conducted by (Öztemel & Şensoy, 2004).
A graph of compressive strength ( fci ), versus the concrete age was plotted for
concrete class, and a logarithmic function was derived from these graphs for each concrete class in order to get an approximate value of a 28-days compressive strength for data of each concrete class. Figures 3 - 7 shows the graph illustrated for each concrete class.
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Figure 3. Compressive Strength Distribution for C16 Data
Figure 4. Compressive Strength Distribution for C20 Data.
y = 30.693ln(x) + 159.79 R² = 0.1104 0 50 100 150 200 250 300 350 400 450 0 5 10 15 20 25 30 35 40 C o m p res si v e S tren g th ( k g /cm 2)
Concrete Age (days)
C16
y = 54.128ln(x) + 130.42 R² = 0.3486 0 100 200 300 400 500 600 0 20 40 60 80 100 120 140 160 C o m p res si v e S tren g th ( k g /cm 2)Concret age (days)
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Figure 5. Compressive Strength Distribution for C25 Data
Figure 6. Compressive Strength Distribution for C30 Data
y = 82.19ln(x) + 94.154 R² = 0.4108 0 100 200 300 400 500 600 700 800 0 20 40 60 80 100 120 140 160 C o m p res si v e S tren g th ( k g /cm 2)
Concrete age (days)
C25
y = 48.093ln(x) + 260.46 R² = 0.2441 0 100 200 300 400 500 600 700 0 20 40 60 80 100 120 140 C o m p res si v e S tren g th ( k g /cm 2)Concrete age (days)
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Figure 7. Compressive Strength Distribution for C30 Data
Utilizing the data from the graph, a logarithmic function is derived to calculate the approximate 28-days compressive strength. Below is the list of the logarithmic function for each concrete class
y=30.693ln(x) + 159.79
y = 54.128ln(x) + 130.42
fci (x)= y = 82.19ln(x) + 94.154
y = 48.093ln(x) + 260.46 y = 47.589ln(x) + 315.32
Where x represents the concrete age in days and fci (x) is the function that depends on
the value of x, which represents the compressive strength corresponding to that concrete age in the particular concrete class.
The approximate 28-days compressive strength is computed using Eq. (3.1) as suggested by (Öztemel & Şensoy, 2004) in their study on compressive strength.
𝑓
𝑐𝑖28(x) = 𝑓𝑐𝑖∗ 𝑓𝑐𝑖(𝑡=28) 𝑓𝑐𝑖(𝑡=𝑡𝑡𝑒𝑠𝑡) (3.1) y = 47.589ln(x) + 315.32 R² = 0.2322 0 100 200 300 400 500 600 700 0 100 200 300 400 500 C o m p res si v e S tren g th ( k g /cm 2)Concrete age (days)
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Here Xtest is therelevant concrete age of testing, 𝑓𝑐𝑖(x) is the compressive strength tested for that age in the results from the laboratory, 𝑓𝑐𝑖28is the approximate 28-days compressive strength.
After a complete analysis of the data and the approximate 28-days compressive strength was derived, a further analysis was carried out on the data to obtain the mean, standard deviation and c.o.v for each concrete class, as shown in Table 1.
Table 1. 28 day Statistical parameters on the compressive strength data
Concrete Class C16 C20 C25 C30 C35 Overall
Concrete age 28 28 28 28 28 - Number of Samples 53 506 1051 3127 968 5705 Fck,cyl(Fck,cub) 16(20) 20(25) 25(30) 30(35) 35(37) - Mean µ (Mpa) 26.2 31.1 368.15 420.52 473.78 408.69 Standard deviation 5.79 5.6 68.89 57.71 63.97 76.5 COV 0.22 0.18 0.19 0.14 0.14 0.19
3.2.2 Uncertainty Analysis of Concrete Compressive Strength
Analysis conducted on the collected data, gave a mean value of the cubic compressive strength of concrete to be 40.87 N/mm2 as an Overall value with a c.o.v. of 19%. The equivalent cylindrical compressive strength can be computed using a conversion factor of 0.83, therefore the cylindrical compressive will be 33.92 N/mm2 (40.87 x 0.83 = 33.92), with a c.o.v of 19%. In the content of this chapter, fc denotes
the mean compressive strength and c.o.v is denoted by δfc. The c.o.v is used to
measure the inherent (basic) variability in the compressive strength of concrete. Considering the common construction conditions in Turkey and North Cyprus, the values taken for this study can be used to represent the whole of North Cyprus. In reality, variation occurs in concrete strength of a structure and the strength specified
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at the design stage. This variation may occur due to the following factors; Variations in the properties of materials.
Proportions of concrete mix.
Variations in mixing. Transporting. Placing and curing methods.
Variations in testing procedures.
Additional uncertainties apart from the inherent variability may be caused by some other factors, will be considered according to international literature since there is no much local information to quantify these factors.
The consideration that the concrete strength in a structure is lower than the strength of cylinder specimen tested in the lab from the same sample of concrete. The deviation may arise due to the following effects;
Curing and placing processes.
Segregation of concrete in deep member. Size and shape.
Stress conditions
Due to the effects of these factors, the in-situ strength of concrete is low compared with the strength measure in the laboratory. In order to consider the difference between the in-situ strength and the laboratory strength, a factor, 𝑁1, has been introduced, where 𝑁1 is the correction factor with a c. o. v denoted by ∆1.
In a study conducted by (Ellingwood & Ang, 1972), they found the correction factor within a range from 0.83 to 0.92, to consider the difference between the in-situ strength and the standard cylindrical strength, with a value of ∆1. to be 0.16, as the
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c.o.v associated with this correction factor. In another study of (Mirza 1979), he suggested the correction factor (𝑁1), to be within the range of 0.74 to 0.94 and the value of ∆1as 0.1. In another study conducted by (Firat, 2007), 𝑁1 was computed as 0.86 and ∆1as 0.13. Here a mean value for the correction factor, 𝑁1, is taken as 0.85 as an average value from previous research, and the c.o.v (∆1), is taken as 0.13 as an average value of the reported values.
Strain effect in concrete due to micro cracking and due to creep is another factor causing decrease in the observed in-situ compressive strength of concrete; this can be corrected by using a factor suggested by (Mirza et al. 1979), in their study, denoted by 𝑁2, with its corresponding c.o.v denoted by ∆2, they suggested a formula to be using in computing the value of 𝑁2 given by;
𝑁2 =0.89(1+0.08 log (R)) (3.2)
Here R is the rate of loading measure in psi/sec, R value used during the test as 0.5 psi/sec which is equivalent to 3.447 KN/m2 /sec, therefor the value of 𝑁2 is computed as 0.89. In a study conducted by (Kömürcü, 1995) he found the value of 𝑁2 as 0.88 with a c.o.v as 0 (i.e. no prediction uncertainty was discovered so ∆2=0).
(Mirza et al. 1979) in their study also suggested that the uncertainty associated with the prediction of 𝑁2 can be neglected. In this study, 𝑁2 will be taken as 0.89 without
considering its prediction uncertainty (i.e. ∆2 =0).
Another factor will be consider which deals with the error that rises due to lack of standard testing method, proper timing , poor calibration of machine and human errors in general. This factor has been suggested by (Kömürcü, 1995), in his study as 𝑁3 with a value of 0.95 as the mean and the prediction uncertainty associated with
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this factor as ∆3 with a value of 0.05. All these factors will be considered in this research.
Therefor the overall mean bias in the compressive strength can be computed as is computed with the formula given in Eq. (3.3)
𝑁𝑓𝑐 = 𝑁1+ 𝑁2+ 𝑁3 (3.3)
𝑁𝑓𝑐 = 0.70 (0.85x0.87x0.95 = 0.70), the in-situ mean of the compressive strength can
be computed with the relationship given in Eq. (3.4) below;
𝑓𝑐 = 𝑁𝑓𝑐 x 𝑓̇𝑐 (3.4) Here 𝑓𝑐 denotes the in-situ value of the compressive strength of concrete, 𝑓̇𝑐 denotes the value of the compressive strength from the cylindrical specimens tested in the laboratory, and 𝑁𝑓𝑐 is the value of the overall mean bias in fc. Therefor the mean
in-situ compressive strength is computed as 𝑓𝑐 = 23.74N/mm2 (0.70 x 33.92 = 23.74), as the mean value of the in-situ compressive strength of concrete. The average of the concrete classes used in this study is found to be 25.2 N/mm2 (C16, C20, C25, C30, C35).The nominal compressive strength of concrete can be computed by dividing the compressive strength with a factor of 1.5 which is a value taken from TS 500 (2000), corresponding to the average value of the concrete class. Therefore the nominal compressive strength is computed as 𝑓̇𝑐 = 16.8 N/mm2 (25.2/1.5 = 16.8),
Thus the ratio of mean to nominal ratio is computed as 𝑓𝑐
𝑓̇𝑐 =
23.74
16.8 = 1.41.
The total variability due to the prediction error related to those three uncertainty sources is computed as∆𝑓𝑐= √0.1322+ 02+ 0.052 = 0.14.
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Therefore the total uncertainty will be computed by combining the prediction uncertainty and the inherent uncertainty, as Ω𝑓𝑐 = √0.192+ 0.142 = 0.24. In a
study conducted by (Firat, 2007), he took total uncertainty as 0.18, also in the study of (Kömürcü and Yücemen, 1996), the value of total uncertainty was computed as 0.21. Here, the value computed is related to the data collected from firms around North Cyprus.
The distribution of the compressive strength of concrete has been taken as normally distributed by researchers that worked on it in the past, for the purpose of this study a program Easy Fit 5.6 was used in the determination of the distribution of data on concrete compressive strength and it was found to follow a normal distribution.
3.3 Reinforcing Steel Bars
North Cyprus does not produce steel. Reinforcing steel bars used in North Cyprus are imported from other countries especially from Turkey.
The trades of these iron and steel products are carried out by individual firms. These firms order the steel products with specifications, standards and size required in the construction industries in North Cyprus. Table 2 shows the standards of some countries related to the reinforcing steel bars.
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Table 2. Reinforcing steel bars required standards according to different countries (from Firat, 2007)
Data used in this study on reinforcing steel bars are collected from firms around North Cyprus.
3.3.1 Analysis of Data
Data collected from individual firms on the yield strength, ultimate strength and elongation of reinforce steel bars, were further tested in the laboratory of Mechanical Engineering Department of Eastern Mediterranean University, the further testing was done in order to check the conformity with the data obtained from the firms.
A total of 3851 specimens of reinforcing steel bars were tested for yield strength, ultimate strength and elongation. The test was carried out in accordance with the specifications of TS 708. Minimum Yield S trength Minimum Ultimate Limit S trength Yield/ Ultimate S trength Minimum Elongation ( N/mm² ) ( N/mm² ) ( % ) 12 III a 420 500 1.1 (Ø8-Ø28) IV a 500 550 1.08 10 (Ø32-Ø50) Gr 40 300 500 - 11 ~ 12 Gr 60 420 620 - 7 ~ 9 Gr 75 520 690 - 6 ~ 7 USA ASTM A615 / A616M (1996) S tandard Country Class Gr 460B 460 - 1.08 14 England Turkey BS 4449 (1997) TS 708 (1996)
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Table 3. Analysis on yield strength for 420(a) reinforce steel bars data
Diameter (mm) Number of Samples Mean Yield strength µ (Mpa) Standard deviation COV 8 366 458.28 52.58 0.11 10 837 469.79 56.87 0.12 12 539 464.05 54.62 0.12 14 605 474.29 60.8 0.13 16 489 493.06 64.26 0.13 18 319 481.36 58.3 0.12 20 283 511.46 50.2 0.1 22 337 494.16 43.4 0.09 24 76 523.97 45.69 0.09 Overall 3851 478.78 58.4 0.12
Table 4. Analysis on Ultimate strength of 420(a) reinforcing steel bars data
Diameter (mm) Number of Samples Mean Yield strength µ (Mpa) Standard deviation COV 8 366 629.26 75.57 0.12 10 837 629.8 61.29 0.097 12 539 629.83 64.11 0.1 14 605 633.64 63.47 0.1 16 489 636.77 64.52 0.1 18 319 639.18 64.5 0.1 20 283 640.58 63.5 0.099 22 337 640.04 62.09 0.097 24 76 640.51 61.9 0.097 Overall 3851 640.52 61.87 0.096
3.3.2 Uncertainty Analysis of Reinforcing Steel Bars
The mechanical properties of reinforcing steel bars such as its strength parameter are the main characteristics used to classify it. Therefore, variation in these properties may be analyzed as an uncertainty associated with this strength parameters (i.e. Yield strength and Ultimate strength). In order to properly quantify these uncertainties, the
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sources of these variations should be considered, the following are some of the factors that cause these variations;
• Effect of bar diameter.
• Change in the strength of the material. • Rate of loading during the test.
• Changes in the cross-sectional area.
• Effect of using a combination of bars belonging to another batch.
In a study conducted by (Mirza and MacGregor, 1976), on variations that occurs in reinforcing steel bars, they suggested that a value of 5% to 8% should be taken as coefficient of variation (c.o.v) for data collected from different producers on individual bar size. (Kömürcü, 1995), found variability range from 1% to 4% for individual bar size, provided that the reinforcing steel bars are from the same manufacturer.
In Turkey the c.o.v ranges from 2% to 7% for individual bar size from the same manufacturer. In this study the inherent variability in reinforcing steel bars yield strength is computed as 12% as the overall of the entire bar sizes collected from the firms (i.e. c.o.v denoted by 𝛿𝑓𝑦= 0.12).
Mirza and MacGregor (1979), pointed out that the yield strength of reinforcing steel bars can be overestimated since the test procedure is performed using a huge amount of strain in structures under static loads. Therefor the effect caused by strain and rate of loading can be corrected by using a factor with an overall mean bias, 𝑁1 = 0.9, (Ellingwood & Ang, 1972). Firat (2014), also took this value as 0.9 in his study on
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reinforcing steel bars. In another study carried out by (Kömürcü, 1995) he suggested that the prediction error ∆1 associated with this bias factor can be neglected. Here, the mean bias factor 𝑁1 is taken as 0.9 and the prediction error ∆1 is neglected.
The upper and lower yielding point of reinforcing steel bars are used to determine its yield strength, the yield strength of reinforcing steel bars is also affected by some factors. A correction factor𝑁2, is introduced in order to deal with the effect of these factors. In the study of (Kömürcü & Yücemen, 1995) the correction factor𝑁2, was taken as 1 with a prediction error ∆2𝑎s 9%. Ellingwood and Ang, (1972), used a value of 5% as the prediction error associated with this correction factor. In this study𝑁2, is taken as 1 and the prediction error ∆2 as 5%, as an average value.
The overall inherent variability in yield strength computed from the data analyzed was found to be 12%. If a structure is built using reinforcing steel bar from a single manufacturer the prediction error is assumed to be 0, data collected for the purpose of this study are not entirely from a single manufacturer, thus a prediction error ∆3 is considered with a value of 6% as an average value between 0 and 12%, with a correction factor𝑁3, as 1. The overall mean bias factor related to the yield strength can be computed as 𝑁𝑓𝑦 = 𝑁1+ 𝑁2+ 𝑁3 = 0.9 (0.9 x 1 x 1= 0.9). The overall prediction error ∆𝑓𝑦= �∆12+ ∆22+ ∆12= √0.052+ 0.062 = 0.078 .
Results from the analysis conducted on data from the test on yield strength, a value of 478.78 N/mm2 is computed as the mean. Considering the factors affecting the yield strength of reinforcing steel bar, the corrected mean yield strength is computed as 𝑓𝑦 = 𝑁𝑓𝑦∗ 𝑓̇𝑦 = 0.9 x 478.78 = 430.90 N/mm2 . The nominal yield strength
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corresponding to 420(a) reinforcing steel bars is 365 N/mm2 (420/1.15= 365), according to TS 500 (2000). Therefore, the mean to nominal ratio is computed as 1.18 (i.e.
𝑓𝑦 𝑓̇𝑦 =
430.90
365 = 1.18). Kömürcü and Yücemen (1996), computed 0.14 as the result of
total prediction error on yield strength. A similar research by (Real et al., 2003) suggested a value between the range of 0.05 to 0.10 as a value of total prediction error in yield strength, Firat, 2007, used a value of 0.09 as the total prediction error. Here the total prediction error ∆𝑓𝑦is computed as 0.078. By combining the total prediction error and the inherent variability, the total uncertainty can be computed as Ω𝑓𝑦 =
√0.122 + 0.0782 = 0.14
Ultimate strength of reinforcing steel bars is almost similar in terms of source variation (Mirza and MacGregor, 1979). The mean ultimate strength in this study is computed as 640.52 N/mm2, with an overall mean bias, 𝑁𝑓𝑢 = 0.9. The corrected value of the mean ultimate strength, 𝑓𝑢= 0.9 x 640.52 = 576.47 N/mm2, a nominal value of 435 N/mm2 (500/1.15=435 N/mm2) as stated in (TS 500, 2000). The ration of mean to nominal of the ultimate strength is computed as 1.33 (𝑓̅u
𝑓𝑢 = 576.47
435 = 1.33). The overall prediction
error related to ultimate strength is computed as ∆𝑓𝑢= √0.052+ 0.062+ 0.22 = 0.078. A value of 0.096 was computed from the statistical analysis on ultimate strength as the overall inherent variability. Therefore the total uncertainty related to the ultimate strength is computed as Ω𝑓𝑢 = √0.0962 + 0.0782 = 0.12
Data used for the purpose of this study were found to follow a normal distribution using a computer program Easy Fit 5.6. Firat, 2007, used normal distribution for the yield strength of reinforcing steel bars, while Kömürcü, 1995 used lognormal distribution.
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3.4 Dimensions
Variation in the dimension of reinforced concrete structural members computed during the design stage and that of the as-built, affects the resistance of the reinforced structure members. These variations are considered in terms of geometrical discrepancies, which occur mostly during the construction stage in the life-cycle of the structure. Geometrical discrepancies depend on the size, shape and quality of form work used during the construction. The method of concreting and vibrating these structural members at the construction stage is regarded as the primary source of the discrepancy (Atadero & Karbhari, 2006).
Unfortunately, local data to be used in quantifying the uncertainty related to dimensions was not found, therefore values used in this section are based on the research carried out in turkey on dimensions of reinforced concrete members. Based on engineering judgment, there are many similarities between the workmanship in Turkey and North Cyprus, given that the most construction workers are from Turkey. Prediction errors related to dimension were quantified based on three likely sources of variability, these sources include;
Dimensional changes with change in different design values. Unfixity of forms.
Difficulties in the direct measurement of effective depth.
Variability caused by changes in different design values can be accounted for with a prediction error of 0.02. A value of 0.02 is also taken as the prediction error associated with unfixity of forms, (Firat, 2007). A total prediction error of ∆𝑏𝑤= √0.022+ 0.022 = 0.03. The variability due to difficulties in direct
35
measurement of the effective depth is quantified with a prediction error of 0.06, this value is suggested by Ellingwood et al. (1980), Kömürcü (1995) and Firat (2014). Therefore, the total prediction error related to effective depth of both beams and columns is computed as∆𝑏𝑑= √0.022+ 0.022+ 0.062 = 0.07
Kömürcü (1995), in his study on dimensions of reinforced concrete structures in Turkey computed a value of 0.03, 0.03 and 0.07 as the prediction error associated with the depth, width and effective depth of beams and columns, respectively. In a similar study, Firat (2007) computed the same values for the prediction error of the depth, width and effective depth of beams and columns. In the study, considering the similarities in workmanship between Turkey and North Cyprus, the prediction error related to the depth, width and effective depth of beams and column is taken as 0.03, 0.03 and 0.07, respectively. The inherent variability associated with the width, depth and effective depth of beams and columns will be computed separately, in accordance with previous researches done in Turkey.
3.4.1 Beam Dimensions
In a recent study conducted by (Firat, 2007) on the dimensions of beam, he computed the inherent variability related to the width, depth and effective depth of beam 0.025, 0.045 and 0.024, the mean to nominal ratio was computed as 0.998, 0.996 and 1 for the depth, width and effective depth of beam. By combining the prediction error and the inherent variability a total variability is computed for the width, depth and effective depth of beam as 0.054, 0.04 and 0.074, respectively.
Yücemen and Gulkan, (1989), and Kömürcü (1995), in their research on beam dimension in Turkey, the computed a value of 1 as the ratio of mean to nominal for the depth, width and effective depth of beam. Here, the total variability is taken as
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0.054, 0.04 and 0.074 and the mean to nominal ratio related to the width, depth and effective depth is taken as 0.998, 0.996 and 1, respectively.
Table 5. Statistical analysis results on column dimensions (Firat, 2007)
Beam Dimension Inherent Variability Prediction Error Mean to nominal ratio Total Variability Width (Wb) 0.045 0.03 0.998 0.054 Depth (db) 0.025 0.03 0.996 0.04 Effective depth (deb) 0.024 0.07 1 0.074 3.4.2 Column Dimension
Unfortunately, data related to the width and depth of column is not available for this study, therefore with regards to the similarities of quality control and workmanship between North Cyprus and Turkey, results of researches carried out on the column dimension in Turkey will be used here. Semih & Firat, (2014), conducted a research on column dimension computed a value of 0.032, 0.024 and 0.025 as the prediction error related to the width, depth and effective depth of column, and a mean to nominal ratio result of 1.02, 1.03 and 1.01 for the width, depth and effective depth, respectively. The total variability is computed by combining the inherent variability and prediction error they were found as 0.044, 0.038 and 0.074.
In a similar study conducted by (Yücemen and Gulkan, 1989), and (Kömürcü, 1995), on column dimension also computed the same value for the prediction error. In this study the same values will be taken as the prediction error and the total variability related to the width, depth and effective depth.
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Table 6. Statistical analysis results on column dimensions (Firat, 2007)
Column Dimension Inherent variability Prediction Error Mean to nominal ratio Total Variability Width (Wc) 0.032 0.03 1.02 0.044 Depth (dc) 0.024 0.03 1.03 0.038 Effective depth (dec) 0.025 0.07 1.01 0.074
3.5. Area of Reinforcing Steel Bars
In the study conducted by (Mirza and MacGregor, 1976) on reinforcing steel bars with sizes ranging from 9.5 mm to 35 mm diameters found a value of 0.97 and 0.024 for the mean to ratio and c.o.v, respectively. In another study, (Firat, 2007) computed a value of 1 as the mean to nominal ratio and a total variability ΩAS, as 0.03. In this study a value of 1 and a total variability of 0.03 is taken solely due to the fact that most reinforcing steel bars used in North Cyprus are imported from Turkey.
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Chapter 4
ANALYSIS OF THE RESISTANCE PARAMETER
CONSIDERING DIFFERENT FAILURE MODES.
4.1 Introduction
In this chapter results computed from the previous chapter (Chapter 3) on the uncertainties related to the resistances variables is utilized in order to analyze reinforced concrete structural members considering different failure mode. The mean to nominal ratio and the total uncertainties are computed for structural members in different failure mode, which would be used in the coming chapters for the purpose of calibration.
4.2 Failure Modes in Reinforced Concrete Beams
Beam like other reinforced concrete structures are monolithic in nature, their main purpose is to carry transvers load which creates flexural moments in the beam and shear forces, beams are also subjected to axial load. Therefore two failure modes will be considered for beams in this study (i.e. flexural and shear failure), since they are the most influential parameters governing the beam design.
4.2.1 Shear Strength in Beams
Shear failure in beams are mostly influence by the size of the member and the ratio of shear span to depth. This type of failure occurs due to in adequate shear reinforcement in that structural member, which can be prevented by providing adequate shear reinforcement for the member to attain its maximum limit state in
39
flexure. Thus, the following condition should be satisfied for the design of structural members in shear.
Vr ≥ Vd (4.1)
Where;
Vr: Shear strength in the beam member.
Vd: Maximum shear force in design (Calculated).
The shear strength of a reinforced concrete member can be defined as a summation of the concrete resistance and the shear reinforcement resistance. Here the concrete reinforcement is denoted by Vc, which is considered as 80% of the cracking shear strength in concrete for safety purpose. The shear reinforcement resistance is denoted by Vw. Eq. (4.2 – 4.4) is used to compute these values as specified in TS500 (2000).
Vc = 0.80(0.65fctdbwdψ) (4.2) ψ = 1 + 0.007Nd Ac (4.3) Vw = ASswfywdd (4.4) Where, Ac : Area of concrete
Nd : Design Axial load
Vc: Cracking shear strength.
Fctd: Design tensile strength of concrete
Asw: Cross-sectional area of one stirrup
fywd: Design yield stress of shear reinforcement
S: Stirrup spacing.
The total shear force resisted by the reinforced concrete beams can be computed as; Vr = ASswfywdd + 0.52fctdbwdψ (4.5)
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Given the above relationship in Eq. (4.5) the mean to nominal ratio of the shear failure of the beam can be computed by dividing the mean and the nominal values of the shear force in Eq. (4.6) below;
Vr Vr′
=
Asw S fywdd+0.52fctdbwdψ Asw′ S′ fywd′ d′+0.52fctd′ bw′ d′ψ′ (4.6)The conformity of the design practice in North Cyprus and that of Turkey made easier to use the result of the analysis carried out on the shear strength in reinforced concrete beam in Turkey by (Firat, 2007). The results of the analysis on shear strength in beams yielded a result of the ratio of mean to nominal value of 1.24. Nowak & Szerszen (2003a) computed a value 1.23. Ellingwood et al., (1980) found a value of 1.09 as the mean to nominal value for the shear force in reinforce concrete beams.
The average total uncertainty was reported to be 0.17 (Firat, 2007). In a similar study conducted by (Ellingwood et al., 1980), reported a value of 0.115 as the total uncertainty value. Nowak & Szerszen (2003a) found a value of 0.11 as the total uncertainty.
Here, the value of mean to nominal and the total uncertainty related to shear strength in beams (ΩVr) are taken as 1.24 and 0.17, respectively.
4.2.2 Flexural Strength of Beam
The effect of flexure in beams is that it creates bending stress in the beam member, if the bending moment is positive it produces tensile stress and compressive strain in the bottom and top of the beam, respectively, the opposite happens when the bending moment turns out to be negative. Therefore, for a beam to be safe it should be able to resist these stresses and strain caused due to flexure. To avoid structural failure,
